Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $z = \dfrac{8t + 12}{-5} \div \dfrac{t(2t + 3)}{4t} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{8t + 12}{-5} \times \dfrac{4t}{t(2t + 3)} $ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ (8t + 12) \times 4t } { -5 \times t(2t + 3) } $ $ z = \dfrac {4t \times 4(2t + 3)} {-5 \times t(2t + 3)} $ $ z = \dfrac{16t(2t + 3)}{-5t(2t + 3)} $ We can cancel the $2t + 3$ so long as $2t + 3 \neq 0$ Therefore $t \neq -\dfrac{3}{2}$ $z = \dfrac{16t \cancel{(2t + 3})}{-5t \cancel{(2t + 3)}} = -\dfrac{16t}{5t} = -\dfrac{16}{5} $